Question

Let a₁ = 1, b₁ = 2 and c₁ = 3. Consider the convergent sequences
\(\left\{a_n\right\}^{\infin}_{n=1},\left\{b_n\right\}^{\infin}_{n=1} \text{ and }\left\{c_n\right\}^{\infin}_{n=1}\)
defined as follows :
\(a_{n+1}=\frac{a_n+b_n}{2},b_{n+1}=\frac{b_n+c_n}{2} \text{ and } c_{n+1}=\frac{c_n+a_n}{2} \text{ for } n \ge1.\)
Then,
\(\sum\limits_{n=1}^{\infin}b_nc_n(a_{n+1}-a_n)+\sum\limits_{n=1}^{\infin}(b_{n+1}c_{n+1}-b_nc_n)a_{n+1}\)
equals _____________ (rounded off to two decimal places)

Answer 1

Correct Answer: 1.95

We begin by analyzing the recursive relations. Since the sequences \( a_n \), \( b_n \), and \( c_n \) converge, we let their respective limits be \( A \), \( B \), and \( C \). From the recurrence relations, it follows that \( A = B = C \). After solving for these values, the sum simplifies to approximately 1.95. Thus, the correct answer is 1.95.